For natural examples of locally small categories like the category $\mathbf{Grp}$ of groups, the isomorphism classes themselves are normally not sets. In a set theory like ZFC, even the collection of a finite number of such isomorphism classes would not be a set, because the members of a set must be sets themselves.

But suppose we had a set theory with ur-elements and classes, which would allow us to form such collections, as long as they are well enough defined. For some categories like the category $\mathbf{FinSet}$ of finite sets, the collection of isomorphism classes will form a nice countable set. I'm not sure what happens for the category of (hereditarily) countable sets. (Maybe the corresponding collection will still be countable?)

For a category like $\mathbf{Grp}$, the collection of isomorphism classes will probably not be countable, but instead will probably be quite "huge" relative to the available set universe. I'm not sure whether it will "always" be a set, but still this collection seems to be much smaller to me than any single isomorphism class of this category. But how can we distinguish a natural category like $\mathbf{Grp}$ where the collection of isomorphism classes is still "reasonably small" from unnatural categories where the notion of isomorphism is too restricted, such that the collection of isomorphism classes cannot be a set, even for the most powerful set theories?

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    $\begingroup$ The collection of isomorphism classes in $\mathbf{FinSet}$ is countable: just identify each set with the number of elements in it! Similarly for the collection of isomorphism classes of countable sets. On the other hand, the collection of isomorphism classes in $\mathbf{Grp}$ has the same "cardinality" as the collection of isomorphism classes in $\mathbf{Set}$, which in turn has the same "cardinality" as the class of all cardinal numbers. $\endgroup$ – Zhen Lin Apr 26 '14 at 12:28

One way to circumvent this kind of issue is to work with Grothendieck's universes introduced in SGA4 (as an appendix by Bourbaki if I remember correctly).

For any universe $\mathbb U$, define $\mathbb U$-categories as categories where the hom-sets are elements of $\mathbb U$. So you have the categories $\mathbb U{-}\mathsf{Set}$, $\mathbb U{-}\mathsf{Grp}$, etc.

Then the most important axiom of Grothendieck universes is : for any naive set, there exists a universe containing it. Apply it to, say, the collection $\operatorname{Ob}(\mathbb U{-}\mathsf{Grp})$ which is in some universe $\mathbb V$ : the category $\mathbb U{-}\mathsf{Grp}$ is then a small $\mathbb V$-category.

In that framework, there is no unnatural category !

  • $\begingroup$ I will have to read the linked page in more detail to see whether this answers my question. However, your claim "In that framework, there is no unnatural category !" seems to rest on your special interpretation of "for any naive set", and I couldn't find a definition of "naive set" on the linked page. Note that for me, an isomorphism class of a typical natural category is as different from a "naive set" as possible, because the defining property of its members is that they are undefined, except for their "isomorphic properties". $\endgroup$ – Thomas Klimpel Apr 26 '14 at 14:41
  • $\begingroup$ @ThomasKlimpel In any set theory, you need some meta set notion. For example, working with ZFC, a model of ZFC is a by definition a structure in the language $\{\in\}$, and a structure need to be some kind of set but cannot be a formal set as it is yet to be defined. In my answer, "naive set" refers to that notion of meta set you have. You can replace it by "collection" I think. $\endgroup$ – Pece Apr 26 '14 at 15:05
  • $\begingroup$ Thanks. I guess the term set is a bit too overloaded. I'm thinking of an "extensionally defined collection" when I hear set, in contrast to an "intensionally defined collection", which I associate more with the word class. A collection on the other hand doesn't even need to be well defined for me. You seem to think of set as a "somehow well defined collection", more or less. Tarski has published two different meta truth definitions, one in 1933 making no reference to any meta set notion, and one model-theoretic truth definition in 1956 (with Robert Vaught) using a meta set notion. $\endgroup$ – Thomas Klimpel Apr 26 '14 at 15:40

While ZF and ZFC are not really expressive enough for even properly stating my question, the conservative extension NBG works pretty well for this question, at least as far as classes are concerned.

  • The isomorphism classes are normal classes in the sense of NBG.
  • In a set theory with ur-elements, it would be nice if it were possible to turn these classes into ur-elements, by forming the corresponding 1-tuples. There are no ur-elements in NBG, but we can select a representant from each isomorphism class instead. This is possible, because we have global choice in NBG.
  • The resulting collection is once again a normal class from NBG, so that we can talk about its size. Because we have limitation of size in NBG, all classes which are not sets have the same size. As pointed out by Zhen Lin, the collection of isomorphism classes of $\mathbf{Grp}$ has the same size as the class of all cardinal numbers. A single isomorphism class can't be bigger than that, which seems to answer my question in the negative sense.
  • Using TG instead of NBG doesn't really change the state of affairs related to this question. Hence even so I learned a bit from Pece's answer, it is ultimatively misleading. This is because the consistency strength of ZF is not really the issue. It's a different sort of expressive power, which is needed here. Being able to talk about classes is one part. Having ur-elements might be the other part (for my intuition), but that would still be a material set theory. This might also be an advantage in a certain sense, but another approach might be to look at a structural set theory. It's sometimes nice to read something new with some concrete questions in mind.

I'm not sure that anybody has really understood my real question. All the answers just concern the set theoretic difficulties, but the question is also tagged as "mathematical-modeling". Yes, I wonder how one can distinguish "correct" modeling of isomorphisms from "evil" equality based modeling. The answers relative to the discussed set theories seem to indicate that it is not possible to distinguish it. However, I'm still not convinced that this is not just an artifact of the reduced ontology of these set theories.

  • $\begingroup$ If I'm understanding your question correctly, I do think you'll be unable to capture evilness with the methods you're using. That's because evil is relative: a property of objects or morphisms of a category is evil just if it's not isomorphism invariant, or roughly equivalently if it uses properties of objects that aren't captured by morphisms. In particular a discrete category in which isomorphism is equality isn't somehow inherently evil: equality isn't the problem, asking for a notion of isomorphism not available to your category is. $\endgroup$ – Kevin Carlson May 1 '14 at 22:45

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